Study of Meshless Method Based on Compactly Supported Radial Basis Functions and Point Interpolation Method

Abstract: Meshless method rapidly developed in recent years is a new numerical analysis method. It abandons the concept of elements or meshes that used in conventional methods such as finite element method (FEM). By this method, only nodal data is required, and no element connectivity is needed. The mesh-free characteristic has distinctive advantages to deal with high-speed impact, extremely large deformations, fracture and fragmentation problems, etc. Meshless method offers a wide application in engineering.The local characteristic of compactly supported radial basis functions (RBFs), which makes the stiffness matrix of the governing equation sparse and banded, is well adopted in meshless methods. However, its accuracy is not much high in interpolation. On the other hand, interpolation functions in the point interpolation method (PIM) have delta function property, and it is convenient to apply the essential boundary conditions. The limitation of the PIM is that the matrix may be singular sometimes. In the paper, a approach to modify the compactly supported RBFs is developed. The modified compactly supported RBFs substitute for the polynomial basis functions to construct interpolation functions, then the compactly supported radial point interpolation method is obtained. The formulated shape functions have the property of delta function and the essential boundary conditions can be applied as easy as in conventional finite element method (FEM). However, the proposed method has to use background cells to complete auxiliary computation. Therefore, the local residual idea is used to construct the local radial point interpolation method. For this method, a group of arbitrarily distributed points is used to represent the problem domain, and the background cells are not required. It is a truly meshless method.After that, the implementation of the compactly supported radial point interpolation method is analyzed in detail especially. The flowchart of the theory and the program is presented. In the meantime, a computer program about the two-dimensional elastic-static plane problem is developed in Visual Fortran, and several typical examples and an engineering problem are analyzed. The numerical results show that the proposed method is accurate, convenient and efficient…
Key words: meshless method; point interpolation method; radial basis functions; compactly supported radial point interpolation method; local radial point interpolation method

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